


Fuzzy Measures

by Bitenomnom



Series: Mathematical Proof [8]
Category: Sherlock (TV)
Genre: Friendship, Gen, John Watson has an appropriate level of appreciation for puns and Sherlock does not, John Watson's Blog, John Watson's belly, M/M, Mathematics, but you may put on your goggles if you wish, not necessarily Johnlock
Language: English
Status: Completed
Published: 2012-09-14
Updated: 2012-09-14
Packaged: 2017-11-14 05:08:18
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,200
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/511646
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>“Honestly, John? ‘<i>The Navel Treatment</i>'?”<br/>“You can call it whatever you want on <i>your</i> blog,” John glanced up from his laptop to Sherlock, who was watching him type about the case over his shoulder.<br/>“I didn’t think you were really going to give it that title.”<br/>“You <i>knew</i> I was going to, Sherlock. You know how I run my blog.”<br/>“Yes: stupidly.”<br/>“And,” John pointedly ignored this comment, “it’s not my fault you found the crucial evidence in the victim’s belly button. It was <i>lint</i>, Sherlock. I don’t exactly have a lot to work with.”<br/>“You could try <i>not</i> titling your entries with terrible puns.”</p>
            </blockquote>





	Fuzzy Measures

**Author's Note:**

> This was slightly too much fun. As I tried to indicate in the tags, this is not inherently Johnlock, but I tagged it as such since I think Johnlock fans in particular may enjoy this and you can certainly read as the beginnings of a relationship. I think. Maybe.
> 
> My professor has done a lot of work with fuzzy measures and related stuff (including publishing several books). He firmly does not like calling it "fuzzy measure," but I thought it was too perfect not to use for this series. Apparently this is a somewhat misleading name because it's not really related to fuzzy set theory? Anyway, I bet you haven't taken a math class where the word "fuzzify" has been used more than it has in mine.

(Summary of some of my class notes.)

The concept of fuzzy measure was developed by M. Sugeno in 1974. It describes a non-additive set function, and since the 1990s the general trend has been to ignore the continuity (i.e. from above or from below). Since the name is somewhat deceptive some authors since have chosen to use “non-additive measure” to describe fuzzy measure. A non-additive set function may be superadditive or subadditive. Superadditive refers to the property where the measure of the union of two sets is greater than or equal to the sum of the individual measures of the sets, whereas subadditivity refers to the instance where the measure of the union of two sets is less than or equal the sum of the individual measures of the sets. For instance, consider a factory with several workers. They will each have their own rates of production. If they work together, they will have different rates of production; however, these are not necessarily the sum of the rates of the two individuals; it could be more or less (for instance, if they work particularly well together—which would correspond with superadditivity—they could get more work done combined than each would individually—or if they work poorly together, they might get less work done together than they would individually, which corresponds with subadditivity).

 

 ***

 

            “Honestly, John? ‘ _The Navel Treatment’_?”

            “You can call it whatever you want on _your_ blog,” John glanced up from his laptop to Sherlock, who was watching him type about the case over his shoulder.

            “I didn’t think you were really going to give it that title.”

            “You _knew_ I was going to, Sherlock. You know how I run my blog.”

            “Yes: stupidly.”

            “And,” John pointedly ignored this comment, “it’s not _my_ fault you found the crucial evidence in the victim’s belly button. It was _lint_ , Sherlock. I don’t exactly have a lot to work with.”

            “You could try _not_ titling your entries with terrible puns.”

            “What gave puns such a bad rap?” John gave up on his typing and closed the lid of his laptop. Better than Sherlock continuing on to point out any of his inevitable typos or ill-chosen words or whatever else.

            “You know the answer to that question. You recently read a book about it, don’t pretend I didn’t notice that it appeared on our shelf last month.1”

            “Yeah, well. What is it you wanted, anyway?” John stood and stretched his arms and legs a bit as he did so. It had been just about time for a break, anyway; for how ridiculous it sounded, the case had actually been rather complicated. And, he thought, to his credit, it would have taken Sherlock even longer to solve it if John hadn’t pointed out that the note the killer left consisted entirely of lyrics from 1970s rock songs. Then again, John thought, _he_ certainly wouldn’t have been able to figure the whole thing out, at least not any more quickly than Scotland Yard. He smiled to himself as he recalled the utter hilarity of the fact that the culmination of Sherlock’s entire fantastic deduction was him pulling back some bloke’s undershirt to reveal a bit of maroon-colored lint. What was better was the fact that the entire room had broken into a smattering of applause at that moment.

            “What’s amusing you so much?” Sherlock lifted an eyebrow, grabbing John’s laptop and opening it again and glancing over the screen. “ _‘Preen’?_ Do I really _preen,_ John?”

            “A roomful of people clapping at lint,” John answered, “and yes. And what are you doing with my laptop?”

            “I need to log some data,” Sherlock strutted to the kitchen and placed his laptop on the table beside a series of petri dishes.

            “Right, and you couldn’t do that on your own computer or anything, because it’s for some reason upstairs in my room.”

            “Exactly.”

            “Only,” John made his way over to Sherlock, attempting to glance into the dishes while still keeping a safe distance away, “what _is_ your laptop doing in my room?”

            “Sitting very still, I imagine.” Sherlock lifted one of the dishes to eye level. “John, did you open a window last night?”

            “What? Oh, yeah. Bloody stuffy in here, it was. Figured you wouldn’t care.”

            “Well, my fungus certainly did. Did you not _see_ that I was performing an experiment?”

            “I try not to see them, actually,” John, growing bolder, leaned over the series of dishes. “Wow, that’s awfully fuzzy.”

            “Precisely,” Sherlock agreed, “but not as much as it _would’ve_ been if—”

            “Yeah, okay, fine,” John maneuvered around Sherlock to the refrigerator and opened it. He retrieved a piece of cheese from a drawer and set it on the table. “I don’t think we’ve a shortage of fuzzy things for you to study, Sherlock.”

            “Indeed,” Sherlock agreed absently, tilting his petri dish to look at its contents from a different angle.

            “Is this _related_ to the Navel—”

            “Don’t _say_ that name. No. It’s not. I started this experiment days before that case.”

            John snatched the cheese back from the table and binned it before returning his attention to the refrigerator. “We’ve nothing in here. I _swear_ I’d saved some of the takeout from last night…”

            “I ate it.”

            “You ate?”

            Sherlock sighed. “I occasionally require caloric input for the continued function of my brain.” He narrowed his eyes and grabbed a loose spoon from the table to prod at the fungus, and was apparently displeased with what he observed, based on his frown as he set the dish down, typed something into a spreadsheet on John’s laptop, and picked up the next.

            “Right. And, you know, not that you care, but I’m sure the rest of your body appreciates it too.”

            “I’m sure you do,” Sherlock mumbled.

            John paused midway through shutting the refrigerator. “I do what?”

            “Appreciate my—oh,” Sherlock paused when he glanced toward John and noted his baffled expression. “That’s not what you said. Never mind.”

            “Right,” John finished closing the door. “Well. Either I’m off to the grocery, or getting takeout again. Anything you want?”

            “Peaches,” Sherlock answered.

            “Yeah, didn’t think—oh. Right. Okay. Peaches. How many?”

            “One or two will be sufficient.”

            “Is this somehow related to your experiment?”

            “Perhaps.” Sherlock paused. “And not the canned sort, naturally.”

            “Okay. One or two fresh peaches. Anything you’d like to, I don’t know, eat?”

            “I ate this morning,” Sherlock answered, as if that was the end of it. John wasn’t especially inclined to argue it just now—at least he actually _had_ eaten something, and actual food at that, rather than a couple of biscuits with tea that was still sitting on the kitchen table, cold and only half finished, at dinnertime.

            “You could probably grow mold in your tea,” John said offhandedly as he checked to make sure Sherlock’s card was in his wallet in case his acted up again. Chip and pin machines had a longstanding vendetta against John Watson and he was most certainly not making two trips out tonight. “It sits out long enough.”

            “Peaches, John,” Sherlock waved his hand limply in John’s direction.

            “Right,” John said, and left.

 

 

 

            When John returned, Sherlock had put a segment of one of his jumpers beneath his microscope. Correction: Sherlock had sliced a segment of one of John’s jumpers _from_ the jumper, and _then_ placed it beneath the microscope. “Have you ever thought about it, John?” he was asking as John set the bags on the counter, making a very valiant effort to not clock Sherlock for not being able to go an _hour_ without taking something of his and ruining it in the name of science. John could only assume that by ‘it’ Sherlock meant ‘why I would be such an inconsiderate arsehole to you without any provocation whatsoever and apparently without the slightest hint of remorse, because Sherlock bloody Holmes doesn’t _do_ remorse’

But it probably wouldn’t be terribly productive to spell that out, so instead he asked, “Thought about what?” while silently mourning the loss of his third-favorite jumper.

“Did you ever do anything relating to detective work prior to our acquaintance?”

            “No.” John placed the peaches on the table next to Sherlock, although he was rather regretting doing Sherlock that favor now that he knew what had become of a staple of his wardrobe while he was out doing it.

            “Obviously. I thought not,” Sherlock answered with a tone that almost certainly meant nothing, but nonetheless left John with a bit of a sting.

            “I mean, diagnosing patients is a bit like solving a…”

            “Yes, every bit as exciting as solving the sort of cases Lestrade gives me, I agree—which is why you joined the army.”

            “Right, okay, point taken. What about it?”

            “I find it curious,” Sherlock looked up from the jumper to pick up one of the peaches, running his thumb over it, “that overall the amount of time it takes me to solve a case has decreased disproportionately to a mere increase in experience, even factoring in the minor amount of help you provide with your knowledge about pop culture and similar trivia—”

            “Like the solar system,” John added. Sherlock’s reaction to that reminder never got old: an eyeroll that was used to poorly mask at once two distinct layers of Sherlock’s actual reaction: frustration with himself and amusement at John’s continual efforts to bring it up.

            “—And even including time saved by overall fewer criminals per person to wrestle, shoot, or otherwise subdue.”

            “You know what I find odd,” John countered.

            “Oddly, no, although given the tone of your voice, your clenched fist, and the direction of your gaze, I suspect it involves my dissection of your clothing.”

            “Somehow, despite the fact that before meeting you on average I had to get a new jumper every few months or half a year or so, and before meeting me you bought approximately zero jumpers—”

            “One,” Sherlock interrupted, and then clarified, “disguise for a case.”

            “Right. But somehow, since I started living with you, _we_ , combined,end up buying me a new jumper about every month because _somebody_ keeps using them to mop up chemicals or in experiments about mold!”

            “‘ _We_ ’?” Sherlock quirked an eyebrow.

            “Yeah, whose credit card do you think I use when I go out and replace them? I sure as hell am not paying for you to grow fungus on my clothes.”

            “I’m not studying _fungus,_ John.” _Obviously,_ said his tone, because _obviously_ the petri dishes full of fungus on their kitchen table were completely unrelated to studying fungus.

            “Oh yeah? Then what, pray tell, are you studying?”

            “Fuzz,” Sherlock tilted his head toward the peaches.

            “Fuzz. Right—because of the Navel—”

            “ _No,_ John. Because,” he paused and picked the peach up again and then set it back down, “because I…” he paused again, his mouth opening and then closing as he appeared to grasp for words unsuccessfully for roughly the first time in his life.

            “Whatever it is you’re worried about saying, rest assured that you cannot piss me off more than you already have by destroying my jumper whilst I go buy produce for you to let sit here and get rotten.”

            “ _John_.” Sherlock stood abruptly. “I have been attempting to replicate the texture of…of…”

            John arched an eyebrow.

            “I may have…”

            John crossed his arms.

            “That is, it was not precisely intentional, except that…”

            John tilted his chin up.

            “Well,” Sherlock breathed in, “you had sort of passed out on the sofa and your shirt had been disrupted by the combination of your gradual unconscious descent down the back of the sofa and the frictional interaction between your clothing and the surface of the sofa, resulting in a slight gap between the upper and lower portions of your clothing, and I was trying to play my violin and it was distracting so I thought you weren’t likely to notice if I simply returned your shirt to its original position, but during that process my knuckles may have, to a limited but not insignificant extent, brushed against your belly. And…it has a unique texture.” Sherlock breathed out the rather impressive amount of his breath that was still left in his lungs after his admission. “And I was simply attempting to replicate it.”

            “By studying fungus, a jumper, and peaches.”

            “And cat fur. Not,” Sherlock held up a hand, “from a living cat. I mean, not from a dead cat either. Fur naturally brushed from an average, untraumatised cat2.”

            “And…”

            “Measures of their fuzziness are inconclusive. And anyway, I only have one brief recollection to work from.”

            “Right,” John raised one hand to his face and massaged his temples. “Right then.” He sighed and curled his hands around the bottom of his current, still-intact (for now) jumper, and pulled it up to reveal his stomach. “There you go. Have at it. Merry Christmas.”

            “It’s August, John,” Sherlock answered, his voice small and earnest, as if its not being the proper month for Christmastime was the only thing giving him pause.

            “Happy birthday, then.”

            “My birthday isn’t in August either.”

            “Get your sodding hand on my sodding stomach, Sherlock. I didn’t buy those peaches for nothing.”

           

 

 

**Author's Note:**

> 1: This book exists. _The Pun Also Rises: How the Humble Pun Revolutionized Language, Changed History, and Made Wordplay More Than Some Antics_ by John Pollack. Rather interesting; I more or less read the entire thing in one sitting.
> 
>  
> 
> 2:          “Did you need something?” Molly squeaked.
> 
>             “What makes you think I need something?”
> 
>             “You bought me some crisps—I know they’re for me. And um, you keep looking at me like you’re trying to find something to compliment in case I say ‘no’ to whatever you’re about to ask.”
> 
>             Sherlock couldn’t restrain a hint of a smirk.
> 
>             “So what is it?”
> 
>             “Cat fur.”
> 
>             “Cat fur?”
> 
>             “That’s what I said. For an experiment.”
> 
>             “What are you…experimenting…about?”
> 
>             “None of your concern. You have a cat, do you not?”
> 
>             “Yeah,” she smiled, “Toby.”  
> 
>             “Well, please collect some of his fur in this,” he handed her a baggie. “And be prompt about it. I shall retrieve it tomorrow.”
> 
>             “I-I’m not going to… _shave_ my cat for your experiment! …Oh. You mean fur from when I brush him.”
> 
>             Sherlock was already halfway out the door, and waved at her dismissively. “Whatever you wish to do, just ensure there is at least half a gram of it in that bag by this time tomorrow.”
> 
>             “Right. I’ll just brush him then,” she said to the closing door.


End file.
